Descartes refused to accept the authority of previous philosophers. He frequently set his views apart from those of his predecessors. In the opening section of the Les passions de l'âme, a treatise on the early modern version of what are now commonly called emotions, Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before". His best known philosophical statement is "Cogito ergo sum" (French: Je pense, donc je suis; I think, therefore I am), found in part IV of Discours de la méthode (1637; written in French but with inclusion of "Cogito ergo sum") and §7 of part I of Principles of Philosophy (1644; written in Latin).[14]

Many elements of his philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into matter and form; second, he rejected any appeal to final ends, divine or natural, in explaining natural phenomena.[15] In his theology, he insists on the absolute freedom of God's act of creation.

Descartes laid the foundation for 17th-century continental rationalism, later advocated by Spinoza and Leibniz, and opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, and Hume. Leibniz, Spinoza[16] and Descartes were all well-versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well.

René du Perron Descartes was born in La Haye en Touraine (now Descartes, Indre-et-Loire), France, on 31 March 1596.[17] His mother, Jeanne Brochard, died soon after giving birth to him, and so he was not expected to survive.[17] Descartes' father, Joachim, was a member of the Parlement of Brittany at Rennes.[18] René lived with his grandmother and with his great-uncle. Although the Descartes family was Roman Catholic, the Poitou region was controlled by the Protestant Huguenots.[19] In 1607, late because of his fragile health, he entered the JesuitCollège Royal Henry-Le-Grand at La Flèche,[20][21] where he was introduced to mathematics and physics, including Galileo's work.[20][22] After graduation in 1614, he studied for two years (1615–16) at the University of Poitiers, earning a Baccalauréat and Licence in canon and civil law in 1616,[20] in accordance with his father's wishes that he should become a lawyer.[23] From there he moved to Paris.

In his book Discourse on the Method, Descartes recalls,

I entirely abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it.

According to Adrien Baillet, on the night of 10–11 November 1619 (St. Martin's Day), while stationed in Neuburg an der Donau, Descartes shut himself in a room with an "oven" (probably a Kachelofen or masonry heater) to escape the cold. While within, he had three dreams[27] and believed that a divine spirit revealed to him a new philosophy. However, it is likely that what Descartes considered to be his second dream was actually an episode of Exploding head syndrome.[28] Upon exiting, he had formulated analytical geometry and the idea of applying the mathematical method to philosophy. He concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life's work.[29][30] Descartes also saw very clearly that all truths were linked with one another so that finding a fundamental truth and proceeding with logic would open the way to all science. Descartes discovered this basic truth quite soon: his famous "I think, therefore I am".[24]

In 1620 Descartes left the army. He visited Basilica della Santa Casa in Loreto, then visited various countries before returning to France, and during the next few years spent time in Paris. It was there that he composed his first essay on method: Regulae ad Directionem Ingenii (Rules for the Direction of the Mind).[24] He arrived in La Haye in 1623, selling all of his property to invest in bonds, which provided a comfortable income for the rest of his life.[31] Descartes was present at the siege of La Rochelle by Cardinal Richelieu in 1627.[32] In the fall of the same year, in the residence of the papal nuncio Guidi di Bagno, where he came with Mersenne and many other scholars to listen to a lecture given by the alchemist Nicolas de Villiers, Sieur de Chandoux on the principles of a supposed new philosophy,[33] Cardinal Bérulle urged him to write an exposition of his own new philosophy in some location beyond the reach of the inquisition.[34]

Unlike many moralists of the time, Descartes was not devoid of passions but rather defended them; he wept upon Francine's death in 1640.[36] "Descartes said that he did not believe that one must refrain from tears to prove oneself a man." Russell Shorto postulated that the experience of fatherhood and losing a child formed a turning point in Descartes' work, changing its focus from medicine to a quest for universal answers.[37]

Despite frequent moves,[38] he wrote all his major work during his 20+ years in the Netherlands, where he managed to revolutionize mathematics and philosophy.[39] In 1633, Galileo was condemned by the Catholic Church, and Descartes abandoned plans to publish Treatise on the World, his work of the previous four years. Nevertheless, in 1637 he published part of this work[40] in three essays: "Les Météores" (The Meteors), "La Dioptrique" (Dioptrics) and "La Géométrie" (Geometry), preceded by an introduction, his famous Discours de la méthode (Discourse on the Method).[40] In it, Descartes lays out four rules of thought, meant to ensure that our knowledge rests upon a firm foundation.

The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

In La Géométrie, Descartes exploited the discoveries he made with Pierre de Fermat, having been able to do so because his paper, Introduction to Loci, was published posthumously in 1679.[41] This later became known as Cartesian Geometry.[41]

Principia philosophiae, 1644

Descartes continued to publish works concerning both mathematics and philosophy for the rest of his life. In 1641 he published a metaphysics work, Meditationes de Prima Philosophia (Meditations on First Philosophy), written in Latin and thus addressed to the learned. It was followed, in 1644, by Principia Philosophiæ (Principles of Philosophy), a kind of synthesis of the Discourse on the Method and Meditations on First Philosophy. In 1643, Cartesian philosophy was condemned at the University of Utrecht, and Descartes was obliged to flee to the Hague, and settled in Egmond-Binnen.

Descartes began (through Alfonso Polloti, an Italian general in Dutch service) a long correspondence with Princess Elisabeth of Bohemia, devoted mainly to moral and psychological subjects. Connected with this correspondence, in 1649 he published Les Passions de l'âme (Passions of the Soul), that he dedicated to the Princess. In 1647, he was awarded a pension by the Louis XIV of France, though it was never paid.[42] A French translation of Principia Philosophiæ, prepared by Abbot Claude Picot, was published in 1647. This edition Descartes also dedicated to Princess Elisabeth. In the preface to the French edition, Descartes praised true philosophy as a means to attain wisdom. He identifies four ordinary sources to reach wisdom and finally says that there is a fifth, better and more secure, consisting in the search for first causes.[43]

By 1649, Descartes had become famous throughout Europe for being one of the continent's greatest philosophers and scientists.[40] That year, Queen Christina of Sweden invited Descartes to her court in to organize a new scientific academy and tutor her in his ideas about love. She was interested in and stimulated Descartes to publish the "Passions of the Soul", a work based on his correspondence with Princess Elisabeth.[44] Descartes accepted, and moved to Sweden in the middle of winter.[45]

Descartes apparently started giving lessons to Queen Christina after her birthday, three times a week, at 5 a.m, in her cold and draughty castle. Soon it became clear they did not like each other; she did not like his mechanical philosophy, nor did he appreciate her interest in Ancient Greek. By 15 January 1650, Descartes had seen Christina only four or five times. On 1 February he contracted pneumonia and died on 11 February.[48] The cause of death was pneumonia according to Chanut, but peripneumonia according to the doctor Van Wullen who was not allowed to bleed him.[49] (The winter seems to have been mild,[50] except for the second half of January which was harsh as described by Descartes himself; however, "this remark was probably intended to be as much Descartes' take on the intellectual climate as it was about the weather."[44])

In 1996 E. Pies, a German scholar, published a book questioning this account, based on a letter by Johann van Wullen, who had been sent by Christina to treat him, something Descartes refused, and more arguments against its veracity have been raised since.[51] Descartes might have been assassinated[52][53] as he asked for an emetic: wine mixed with tobacco.[54][dubious– discuss]

Initially, Descartes arrives at only a single principle: thought exists. Thought cannot be separated from me, therefore, I exist (Discourse on the Method and Principles of Philosophy). Most famously, this is known as cogito ergo sum (English: "I think, therefore I am"). Therefore, Descartes concluded, if he doubted, then something or someone must be doing the doubting, therefore the very fact that he doubted proved his existence. "The simple meaning of the phrase is that if one is skeptical of existence, that is in and of itself proof that he does exist."[61]

Descartes concludes that he can be certain that he exists because he thinks. But in what form? He perceives his body through the use of the senses; however, these have previously been unreliable. So Descartes determines that the only indubitable knowledge is that he is a thinking thing. Thinking is what he does, and his power must come from his essence. Descartes defines "thought" (cogitatio) as "what happens in me such that I am immediately conscious of it, insofar as I am conscious of it". Thinking is thus every activity of a person of which the person is immediately conscious.[62] He gave reasons for thinking that waking thoughts are distinguishable from dreams, and that one's mind cannot have been "hijacked" by an evil demon placing an illusory external world before one's senses.[63]

And so something that I thought I was seeing with my eyes is in fact grasped solely by the faculty of judgment which is in my mind.

In this manner, Descartes proceeds to construct a system of knowledge, discarding perception as unreliable and, instead, admitting only deduction as a method.[64]

Descartes, influenced by the automatons on display throughout the city of Paris, began to investigate the connection between the mind and body, and how the two interact.[65] His main influences for dualism were theology and physics.[66] The theory on the dualism of mind and body is Descartes' signature doctrine and permeates other theories he advanced. Known as Cartesian dualism, his theory on the separation between the mind and the body went on to influence subsequent Western philosophies. In Meditations on First Philosophy Descartes attempted to demonstrate the existence of God and the distinction between the human soul and the body. Humans are a union of mind and body,[67] thus Descartes' dualism embraced the idea that mind and body are distinct but closely joined. While many contemporary readers of Descartes found the distinction between mind and body difficult to grasp, he thought it was entirely straightforward. Descartes employed the concept of modes, which are the ways in which substances exist. In Principles of Philosophy Descartes explained "we can clearly perceive a substance apart from the mode which we say differs from it, whereas we cannot, conversely, understand the mode apart from the substance". To perceive a mode apart from its substance requires an intellectual abstraction,[68] which Descartes explained as follows:

"The intellectual abstraction consists in my turning my thought away from one part of the contents of this richer idea the better to apply it to the other part with greater attention. Thus, when I consider a shape without thinking of the substance or the extension whose shape it is, I make a mental abstraction."[68]

According to Descartes two substances are really distinct when each of them can exist apart from the other. Thus Descartes reasoned that God is distinct from humans, and the body and mind of a human are also distinct from one another.[69] He argued that the great differences between body and mind make the two always divisible. But that the mind was utterly indivisible, because "when I consider the mind, or myself in so far as I am merely a thinking thing, I am unable to distinguish any part within myself; I understand myself to be something quite single and complete."[70]

In Meditations Descartes discussed a piece of wax and exposed the single most characteristic doctrine of Cartesian dualism: that the universe contained two radically different kinds of substances - the mind or soul defined as thinking, and the body defined as matter and unthinking.[71] The Aristotelian philosophy of Descartes' days held that the universe was inherently purposeful or theological. Everything that happened, be it the motion of the stars or the growth of a tree, was supposedly explainable by a certain purpose, goal or end that worked its way out within nature. Aristotle called this the "final cause", and these final causes were indispensable for explaining the ways nature operated. With his theory on dualism Descartes fired the opening shot for the battle between the traditional Aristotelian science and the new science of Kepler and Galileo which denied the final cause for explaining nature. Descartes' dualism provided the philosophical rationale for the latter and he expelled the final cause from the physical universe (or res extensa). For Descartes the only place left for the final cause was the mind (or res cogitans). Therefore, while Cartesian dualism paved the way for modern physics, it also held the door open for religious beliefs about the immortality of the soul.[72]

Descartes' dualism of mind and matter implied a concept of human beings. A human was according to Descartes a composite entity of mind and body. Descartes gave priority to the mind and argued that the mind could exist without the body, but the body could not exist without the mind. In Meditations Descartes even argues that while the mind is a substance, the body is composed only of "accidents".[73] But he did argue that mind and body are closely joined,[74] because:

"Nature also teaches me, by the sensations of pain, hunger, thirst and so on, that I am not merely present in my body as a pilot in his ship, but that I am very closely joined and, as it were, intermingled with it, so that I and the body form a unit. If this were not so, I, who am nothing but a thinking thing, would not feel pain when the body was hurt, but would perceive the damage purely by the intellect, just as a sailor perceives by sight if anything in his ship is broken.[74]

Descartes' discussion on embodiment raised one of the most perplexing problems of his dualism philosophy: What exactly is the relationship of union between the mind and the body of a person?[74] Therefore, Cartesian dualism set the agenda for philosophical discussion of the mind–body problem for many years after Descartes' death.[75] Descartes was also a rationalist and believed in the power of innate ideas.[76] Descartes argued the theory of innate knowledge and that all humans were born with knowledge through the higher power of God. It was this theory of innate knowledge that later led philosopher John Locke (1632-1704) to combat the theory of empiricism, which held that all knowledge is acquired through experience.[77]

In The Passions of the Soul written between 1645 and 1646 Descartes discussed the common contemporary belief that the human body contained animal spirits. These animal spirits were believed to be light and roaming fluids circulating rapidly around the nervous system between the brain and the muscles, and served as a metaphor for feelings, like being in high or bad spirit. These animal spirits were believed to affect the human soul, or passions of the soul. Descartes distinguished six basic passions: wonder, love, hatred, desire, joy and sadness. All of these passions, he argued, represented different combinations of the original spirit, and influenced the soul to will or want certain actions. He argued, for example, that fear is a passion that moves the soul to generate a response in the body. In line with his dualist teachings on the separation between the soul and the body, he hypothesized that some part of the brain served as a connector between the soul and the body and singled out the pineal gland as connector.[78] Descartes argued that signals passed from the ear and the eye to the pineal gland, through animal spirits. Thus different motions in the gland cause various animal spirits. He argued that these motions in the pineal gland are based on God's will and that humans are supposed to want and like things that are useful to them. But he also argued that the animal spirits that moved around the body could distort the commands from the pineal gland, thus humans had to learn how to control their passions.[79]

Descartes advanced a theory on automatic bodily reactions to external events which influenced 19th century reflex theory. He argued that external motions such as touch and sound reach the endings of the nerves and affect the animal spirits. Heat from fire affects a spot on the skin and sets in motion a chain of reactions, with the animal spirits reaching the brain through the central nervous system, and in turn animal spirits are sent back to the muscles to move the hand away from the fire.[79] Through this chain of reactions the automatic reactions of the body do not require a thought process.[76]

Above all he was among the first scientists who believed that the soul should be subject to scientific investigation. He challenged the views of his contemporaries that the soul was divine, thus religious authorities regarded his books as dangerous. Descartes' writings went on to form the basis for theories on emotions and how cognitive evaluations were translated into affective processes. Descartes believed that the brain resembled a working machine and unlike many of his contemporaries believed that mathematics and mechanics could explain the most complicated processes of the mind. In the 20th century Alan Turing advanced computer science based on mathematical biology as inspired by Descartes. His theories on reflexes also served as the foundation for advanced physiological theories more than 200 years after his death. The Nobel Prize winning physiologist Ivan Pavlov was a great admirer of Descartes.[80]

There are three kinds of ideas, Descartes explained: Fabricated, Innate and Adventitious. Fabricated ideas are inventions made by the mind. For example, a person has never eaten moose but assumes it tastes like cow. Adventitious ideas are ideas that cannot be manipulated or changed by the mind. For example, a person stands in a cold room, they can only think of the feeling as cold and nothing else. Innate ideas are set ideas made by God in a person’s mind. For example, the features of a shape can be examined and set aside, but its content can never be manipulated to cause it not to be a three sided object.[81]

For Descartes, ethics was a science, the highest and most perfect of them. Like the rest of the sciences, ethics had its roots in metaphysics.[64] In this way, he argues for the existence of God, investigates the place of man in nature, formulates the theory of mind-body dualism, and defends free will. However, as he was a convinced rationalist, Descartes clearly states that reason is sufficient in the search for the goods that we should seek, and virtue consists in the correct reasoning that should guide our actions. Nevertheless, the quality of this reasoning depends on knowledge, because a well-informed mind will be more capable of making good choices, and it also depends on mental condition. For this reason, he said that a complete moral philosophy should include the study of the body. He discussed this subject in the correspondence with Princess Elisabeth of Bohemia, and as a result wrote his work The Passions of the Soul, that contains a study of the psychosomatic processes and reactions in man, with an emphasis on emotions or passions.[82] His works about human passion and emotion would be the basis for the philosophy of his followers, (see Cartesianism), and would have a lasting impact on ideas concerning what literature and art should be, specifically how it should invoke emotion.[83]

Humans should seek the sovereign good that Descartes, following Zeno, identifies with virtue, as this produces a solid blessedness or pleasure. For Epicurus the sovereign good was pleasure, and Descartes says that, in fact, this is not in contradiction with Zeno's teaching, because virtue produces a spiritual pleasure, that is better than bodily pleasure. Regarding Aristotle's opinion that happiness depends on the goods of fortune, Descartes does not deny that this good contributes to happiness but remarks that they are in great proportion outside one's own control, whereas one's mind is under one's complete control.[82] The moral writings of Descartes came at the last part of his life, but earlier, in his Discourse on the Method he adopted three maxims to be able to act while he put all his ideas into doubt. This is known as his "Provisional Morals".

In the third and fifth Meditation, he offers an ontological proof of a benevolent God (through both the ontological argument and trademark argument). Because God is benevolent, he can have some faith in the account of reality his senses provide him, for God has provided him with a working mind and sensory system and does not desire to deceive him. From this supposition, however, he finally establishes the possibility of acquiring knowledge about the world based on deduction and perception. Regarding epistemology, therefore, he can be said to have contributed such ideas as a rigorous conception of foundationalism and the possibility that reason is the only reliable method of attaining knowledge. He, nevertheless, was very much aware that experimentation was necessary to verify and validate theories.[64]

In his Meditations on First Philosophy Descartes sets forth two proofs for God's existence. One of these is founded upon the possibility of thinking the "idea of a being that is supremely perfect and infinite," and suggests that "of all the ideas that are in me, the idea that I have of God is the most true, the most clear and distinct."[84] Descartes considered himself to be a devout Catholic[55][56][57] and one of the purposes of the Meditations was to defend the Catholic faith. His attempt to ground theological beliefs on reason encountered intense opposition in his time, however: Pascal regarded Descartes' views as rationalist and mechanist, and accused him of deism: "I cannot forgive Descartes; in all his philosophy, Descartes did his best to dispense with God. But Descartes could not avoid prodding God to set the world in motion with a snap of his lordly fingers; after that, he had no more use for God," while a powerful contemporary, Martin Schoock, accused him of atheist beliefs, though Descartes had provided an explicit critique of atheism in his Meditations. The Catholic Church prohibited his books in 1663.[42][85]

Descartes also wrote a response to External world scepticism. Through this method of scepticism, he does not doubt for the sake of doubting but to achieve concrete and reliable information. In other words, certainty. He argues that sensory perceptions come to him involuntarily, and are not willed by him. They are external to his senses, and according to Descartes, this is evidence of the existence of something outside of his mind, and thus, an external world. Descartes goes on to show that the things in the external world are material by arguing that God would not deceive him as to the ideas that are being transmitted, and that God has given him the "propensity" to believe that such ideas are caused by material things. Descartes also believes a substance is something that does not need any assistance to function or exist. Descartes further explains how only God can be a true “substance”. But minds are substances, meaning they need only God for it to function. The mind is a thinking substance. The means for a thinking substance stem from ideas.[81]

Descartes is often regarded as the first thinker to emphasize the use of reason to develop the natural sciences.[86] For him the philosophy was a thinking system that embodied all knowledge, and expressed it in this way:[64]

Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principals, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.

In his Discourse on the Method, he attempts to arrive at a fundamental set of principles that one can know as true without any doubt. To achieve this, he employs a method called hyperbolical/metaphysical doubt, also sometimes referred to as methodological scepticism: he rejects any ideas that can be doubted and then re-establishes them in order to acquire a firm foundation for genuine knowledge.[87] Descartes built his ideas from scratch. He relates this to architecture: the top soil is taken away to create a new building or structure. Descartes calls his doubt the soil and new knowledge the buildings. To Descartes, Aristotle’s foundationalism is incomplete and his method of doubt enhances foundationalism.[63]

Descartes denied that animals had reason or intelligence. He argued that animals did not lack sensations or perceptions, but these could be explained mechanistically.[88] Whereas humans had a soul, or mind, and were able to feel pain and anxiety, animals by virtue of not having a soul could not feel pain or anxiety. If animals showed signs of distress then this was to protect the body from damage, but the innate state needed for them to suffer was absent. Although Descartes' views were not universally accepted they became prominent in Europe and North America, allowing humans to treat animals with impunity. The view that animals were quite separate from humanity and merely machines allowed for the maltreatment of animals, and was sanctioned in law and societal norms until the middle of the 19th century. The publications of Charles Darwin would eventually erode the Cartesian view of animals. Darwin argued that the continuity between humans and other species opened the possibilities that animals did not have dissimilar properties to suffer.[89]

Descartes has often been dubbed the father of modern Western philosophy, the thinker whose approach has profoundly changed the course of Western philosophy and set the basis for modernity.[11][90] The first two of his Meditations on First Philosophy, those that formulate the famous methodic doubt, represent the portion of Descartes' writings that most influenced modern thinking.[91] It has been argued that Descartes himself didn't realize the extent of this revolutionary move.[92] In shifting the debate from "what is true" to "of what can I be certain?," Descartes arguably shifted the authoritative guarantor of truth from God to humanity (even though Descartes himself claimed he received his visions from God) – while the traditional concept of "truth" implies an external authority, "certainty" instead relies on the judgment of the individual.

In an anthropocentric revolution, the human being is now raised to the level of a subject, an agent, an emancipated being equipped with autonomous reason. This was a revolutionary step that established the basis of modernity, the repercussions of which are still being felt: the emancipation of humanity from Christian revelational truth and Church doctrine; humanity making its own law and taking its own stand.[93][94][95] In modernity, the guarantor of truth is not God anymore but human beings, each of whom is a "self-conscious shaper and guarantor" of their own reality.[96][97] In that way, each person is turned into a reasoning adult, a subject and agent,[96] as opposed to a child obedient to God. This change in perspective was characteristic of the shift from the Christian medieval period to the modern period, a shift that had been anticipated in other fields, and which was now being formulated in the field of philosophy by Descartes.[96][98]

This anthropocentric perspective of Descartes' work, establishing human reason as autonomous, provided the basis for the Enlightenment's emancipation from God and the Church. According to Martin Heidegger, the perspective of Descartes' work also provided the basis for all subsequent anthropology.[99] Descartes' philosophical revolution is sometimes said to have sparked modern anthropocentrism and subjectivism.[11][100][101][102]

One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 4 used in x4 to indicate squaring of squaring.[103][104] He was first to assign a fundamental place for algebra in our system of knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such as Pacioli, Cardan, Tartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three-dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity a2 could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area. Although Descartes did not pursue the subject, he preceded Gottfried Wilhelm Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.[105]

Descartes' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics.[106] His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial.

Descartes discovered an early form of the law of conservation of mechanical momentum (a measure of the motion of an object), and envisioned it as pertaining to motion in a straight line, as opposed to perfect circular motion, as Galileo had envisioned it. He outlined his views on the universe in his Principles of Philosophy.

Descartes also made contributions to the field of optics. He showed by using geometric construction and the law of refraction (also known as Descartes' law or more commonly Snell's law) that the angular radius of a rainbow is 42 degrees (i.e., the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°).[107] He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.[108]

Current opinion is that Descartes had the most influence of anyone on the young Newton, and this is arguably one of Descartes' most important contributions. Newton continued Descartes' work on cubic equations, which freed the subject from the fetters of the Greek perspectives. The most important concept was his very modern treatment of independent variables.[109]

Although Descartes was well known in academic circles towards the end of his life, the teaching of his works in schools was controversial. Henri de Roy (Henricus Regius, 1598–1679), Professor of Medicine at the University of Utrecht, was condemned by the Rector of the University, Gijsbert Voet (Voetius), for teaching Descartes' physics.[110]

1618. Musicae Compendium. A treatise on music theory and the aesthetics of music written for Descartes' early collaborator, Isaac Beeckman (first posthumous edition 1650).

1626–1628. Regulae ad directionem ingenii (Rules for the Direction of the Mind). Incomplete. First published posthumously in Dutch translation in 1684 and in the original Latin at Amsterdam in 1701 (R. Des-Cartes Opuscula Posthuma Physica et Mathematica). The best critical edition, which includes the Dutch translation of 1684, is edited by Giovanni Crapulli (The Hague: Martinus Nijhoff, 1966).

1630–1633. Le Monde (The World) and L'Homme (Man). Descartes' first systematic presentation of his natural philosophy. Man was published posthumously in Latin translation in 1662; and The World posthumously in 1664.

1637. Discours de la méthode (Discourse on the Method). An introduction to the Essais, which include the Dioptrique, the Météores and the Géométrie.

1637. La Géométrie (Geometry). Descartes' major work in mathematics. There is an English translation by Michael Mahoney (New York: Dover, 1979).

1641. Meditationes de prima philosophia (Meditations on First Philosophy), also known as Metaphysical Meditations. In Latin; a second edition, published the following year, included an additional objection and reply, and a Letter to Dinet. A French translation by the Duke of Luynes, probably done without Descartes' supervision, was published in 1647. Includes six Objections and Replies.

1644. Principia philosophiae (Principles of Philosophy), a Latin textbook at first intended by Descartes to replace the Aristotelian textbooks then used in universities. A French translation, Principes de philosophie by Claude Picot, under the supervision of Descartes, appeared in 1647 with a letter-preface to Princess Elisabeth of Bohemia.

1648. Responsiones Renati Des Cartes... (Conversation with Burman). Notes on a Q&A session between Descartes and Frans Burman on 16 April 1648. Rediscovered in 1895 and published for the first time in 1896. An annotated bilingual edition (Latin with French translation), edited by Jean-Marie Beyssade, was published in 1981 (Paris: PUF).

1657. Correspondance (three volumes: 1657, 1659, 1667). Published by Descartes' literary executor Claude Clerselier. The third edition, in 1667, was the most complete; Clerselier omitted, however, much of the material pertaining to mathematics.

In January 2010, a previously unknown letter from Descartes, dated 27 May 1641, was found by the Dutch philosopher Erik-Jan Bos when browsing through Google. Bos found the letter mentioned in a summary of autographs kept by Haverford College in Haverford, Pennsylvania. The College was unaware that the letter had never been published. This was the third letter by Descartes found in the last 25 years.[111][112]

^H. Ben-Yami, Descartes' Philosophical Revolution: A Reassessment, Palgrave Macmillan, 2015, p. 179: "[Descartes'] work in mathematics was apparently influenced by Vieta's, despite his denial of any acquaintance with the latter’s work."

^Porter, Roy (1999) [1997]. "The New Science". The Greatest Benefit to Mankind: A Medical History of Humanity from Antiquity to the Present (paperback edition, 135798642 ed.). Great Britain: Harper Collins. p. 217. ISBN0006374549.

The Descartes most familiar to twentieth-century philosophers is the Descartes of the first two Meditations, someone proccupied with hyperbolic doubt of the material world and the certainty of knowledge of the self that emerges from the famous cogito argument.

Husserl has taken Descartes very seriously in a historical as well as in a systematic sense [...] [in The Crisis of the European Sciences and Transcendental Phenomenology, Husserl] finds in the first two Meditations of Descartes a depth which it is difficult to fathom, and which Descartes himself was so little able to appreciate that he let go "the great discovery" he had in his hands.

^Martin Heidegger [1938] (2002) The Age of the World Picture quotation:

For up to Descartes...a particular sub-iectum...lies at the foundation of its own fixed qualities and changing circumstances. The superiority of a sub-iectum...arises out of the claim of man to a...self-supported, unshakeable foundation of truth, in the sense of certainty. Why and how does this claim acquire its decisive authority? The claim originates in that emancipation of man in which he frees himself from obligation to Christian revelational truth and Church doctrine to a legislating for himself that takes its stand upon itself.

When, with the beginning of modern times, religious belief was becoming more and more externalized as a lifeless convention, men of intellect were lifted by a new belief: their great belief in an autonomous philosophy and science. [...] in philosophy, the Meditations were epoch-making in a quite unique sense, and precisely because of their going back to the pure ego cogito. Descartes work has been used, in fact to inaugurates an entirely new kind of philosophy. Changing its total style, philosophy takes a radical turn: from naïve objectivism to transcendental subjectivism.

Oeuvres de Descartes edited by Charles Adam and Paul Tannery, Paris: Léopold Cerf, 1897–1913, 13 volumes; new revised edition, Paris: Vrin-CNRS, 1964–1974, 11 volumes (the first 5 volumes contains the correspondence). [This edition is traditionally cited with the initials AT (for Adam and Tannery) followed by a volume number in Roman numerals; thus AT VII refers to Oeuvres de Descartes volume 7.]

1955. The Philosophical Works, E.S. Haldane and G.R.T. Ross, trans. Dover Publications. This work is traditionally cited with the initials HR (for Haldane and Ross) followed by a volume number in Roman numerals; thus HR II refers to volume 2 of this edition.

1988. The Philosophical Writings of Descartes in 3 vols. Cottingham, J., Stoothoff, R., Kenny, A., and Murdoch, D., trans. Cambridge University Press. This work is traditionally cited with the initials CSM (for Cottingham, Stoothoff, and Murdoch) or CSMK (for Cottingham, Stoothoff, Murdoch, and Kenny) followed by a volume number in Roman numeral; thus CSM II refers to volume 2 of this edition.

1998. René Descartes: The World and Other Writings. Translated and edited by Stephen Gaukroger. Cambridge University Press. (This consists mainly of scientific writings, on physics, biology, astronomy, optics, etc., which were very influential in the 17th and 18th centuries, but which are routinely omitted or much abridged in modern collections of Descartes' philosophical works.)

1628. Regulae ad directionem ingenii. Rules for the Direction of the Natural Intelligence. A Bilingual Edition of the Cartesian Treatise on Method, ed. and tr. by G. Heffernan, Amsterdam-Atlanta: Rodopi, 1998.

1637. The Geometry of René Descartes, tr. by David E. Smith and M. L. Lantham, New York: Dover, 1954.

1641. Meditations on First Philosophy, tr. by J. Cottingham, Cambridge: Cambridge University Press, 1996. Latin original. Alternative English title: Metaphysical Meditations. Includes six Objections and Replies. A second edition published the following year, includes an additional Objection and Reply and a Letter to Dinet. HTML Online Latin-French-English Edition.

Chomsky, Noam (1966). Cartesian Linguistics: A Chapter in the History of Rationalist Thought. New York: Harper & Row. ISBN1-877275-34-4.; (2009) Third Edition, edited with a new introduction by James McGilvray, Cambridge University Press, ISBN978-0-521-70817-3

Watson, Richard A. (2007). Cogito, Ergo Sum: a life of René Descartes. David R Godine. 2002, reprint 2007. ISBN978-1-56792-335-3. Was chosen by the New York Public library as one of "25 Books to Remember from 2002"